Topological orders in 2+1 dimensions are captured by modular tensor categories (MTCs). We propose a correspondence that assigns a fusion category to a pair (M,G), where M is a Seifert 3-manifold and G is an ADE Lie group. We conjecture that the fusion category associated to (M,G) is an MTC if and only if M has trivial first homology group with coefficients in the center of G. The construction determines the spins of anyons and their S-matrix, and provides a constructive way to access the R- and F-symbols from simple building blocks. We explore the possibility that this correspondence provides an alternative classification of MTCs, which is put to the test by realizing all MTCs (unitary or non-unitary) with rank at most 5.

A large class of 4d SCFTs can be engineered by wrapping a stack of M5-branes on a compact space, possibly with defects. ‘t Hooft anomalies are crucial observables for such theories, which often do not admit any known Lagrangian description. Building on the seminal work of Freed, Harvey, Minasian, Moore, we develop systematic tools for extracting the ‘t Hooft anomalies of a geometrically engineered 4d theory using anomaly inflow from the M-theory bulk. We exemplify our tools by studying a class of setups with M5-branes probing a C^2/Z_2 singularity. We argue that these setups define 4d SCFTs which are dual to a class of AdS_5 solutions­—first discussed by Gauntlett, Martelli, Sparks, Waldram—whose field theory interpretation has been a longstanding puzzle.